Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain Ω ⊂ R^N , where the bang–bang weight equals a positive constant m on a ball B ⊂ Ω and a negative constant −m on Ω \ B. The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher–KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of B in Ω. We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of B vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from ∂Ω.